Further examples of GGC and HCM densities

Transcription

Submitted to the Bernoulli
Further examples of GGC and HCM densities
WISSEM JEDIDI and THOMAS SIMON
Département de Mathématiques, Faculté des Sciences, Campus Universitaire 2092, Université
de Tunis - El Manar, Tunisia. E-mail: [email protected]
Laboratoire Paul Painlevé, Université Lille 1, Cité Scientifique, 59655 Villeneuve d’Ascq
Cedex, France. E-mail: [email protected]
We display several examples of generalized gamma convoluted and hyperbolically completely
monotone random variables related to positive α−stable laws. We also obtain new factorizations
for the latter, refining Kanter’s and Pestana-Shanbhag-Sreehari’s. These results give stronger
credit to Bondesson’s hypothesis that positive α−stable densities are hyperbolically completely
monotone whenever α ≤ 1/2.
Keywords: Generalized Gamma convolution, Hyperbolically completely monotone, Hyperbolically monotone, Positive stable density.
1. Introduction
A positive random variable X is called a Generalized Gamma Convolution (GGC) if its
Laplace transform reads
Z ∞
ϕ(x)
E[e−λX ] = exp − aλ +
(1 − e−λx )
dx , λ ≥ 0,
(1.1)
x
0
where a ≥ 0 and ϕ is a completely monotone (CM) function over (0, +∞). The denomination comes from the fact that the above class can be identified as the closure for
weak convergence of finite convolutions of Gamma distributions. We refer to Bondesson
(1992) and Steutel and Van Harn (2003) for comprehensive monographs on such random
variables. From their definition GGC random variables are self-decomposable (SD) hence
infinitely divisible (ID), absolutely continuous and unimodal - see e.g. Sato (1999) for the
proofs of the latter properties. We also see from (1.1) that GGC random variables are
characterized up to translation by the positive Radon measure on (0, +∞) uniquely associated to the CM function ϕ by Bernstein’s theorem, which is called the Thorin measure
of X and whose total mass, ϕ(0+), might be infinite. As an illustration of this characterization, Theorem 4.1.4 in Bondesson (1992) shows that the density of X vanishes in
a+ if ϕ(0+) > 1, whereas it is infinite in a+ if ϕ(0+) < 1. We refer to James, Roynette
and Yor (2008) for a recent survey on GGC variables having a finite Thorin measure,
dealing in particular with their Wiener-Gamma representations and their relations with
Dirichlet processes.
1
imsart-bj ver. 2011/12/06 file: BEJ1104-016.tex date: March 19, 2012
2
W. Jedidi and T. Simon
A positive random variable X is said to be hyperbolically completely monotone (HCM)
if it has a density f on (0, +∞) such that for every u > 0 the function
w = v + 1/v ≥ 2,
Hu (w) = f (uv)f (u/v),
is CM in the variable w (it is easy to see that Hu is always a function of w). In general, a
function f : (0, +∞) → (0, +∞) is said to be HCM when the above CM property holds
for Hu , and this extended definition will be important in the sequel. HCM densities turn
out to be characterized as pointwise limits of densities of the form
x 7→ Cxβ−1
N
Y
(x + yi )−γi
(1.2)
i=1
where all above parameters are positive - see Sections 5.2 and 5.3 in Bondesson (1992).
This characterization yields many explicit examples of GGC random variables, since it is
also true that HCM random variables are GGC - see Theorem 5.2.1 in Bondesson (1992).
Actually, HCM variables appear as a kind of center for GGC in view of Theorem 6.2.1 in
Bondesson (1992) which states that the independent product or quotient of a GGC by a
HCM variable is still a GGC. The HCM class is also stable by independent multiplication
and power transformations of absolute value greater than one. We refer to Bondesson
(1992) for many other properties of HCM densities and functions.
The HCM property is connected to log-concavity in the following way. A positive
random variable X is said to be hyperbolically monotone (HM) if it has a density f on
(0, +∞) such that the above function Hu is non-increasing in the variable w. Similarly as
above, one can extend the HM property to all positive functions on (0, +∞). Obviously,
HCM is a subclass of HM. It is easy to see - see Bondesson (1992) pp. 101-102 - that X is
HM iff its density f is such that t 7→ f (et ) is log-concave on R. This shows that f is a.e.
differentiable with x 7→ xf 0 (x)/f (x) a non-increasing function, so that f 0 has at most
one change of sign and X is unimodal. The main theorem in Cuculescu and Theodorescu
(1998) shows that HM variables are actually multiplicatively strong unimodal, viz. their
independent product with any unimodal random variable is unimodal. From the logconcavity characterization, the HM property is stable by power transformation of any
√
value, and this entails that the inclusion HCM ⊂ HM is strict: if L ∼ Exp(1), then L
is HM but not ID, hence not HCM. From Prékopa’s theorem, the HM property is also
stable by independent multiplication.
For a positive random variable with density, the standard way to derive the GGC
property is to read it from the Laplace transform. For example, it is straightforward
to see that positive α−stable variables are all GGC - see example 3.2.1 in Bondesson
(1992). On the other hand, it is easier to study the HCM property from features on the
density itself and Laplace transforms are barely helpful. As an illustration of this, we show
without much effort in Section 4 of the present paper that the quotient of two positive
α−stable variables, whose density is explicit, is HCM iff α ≤ 1/2. Problems become
usually more intricate when one searches for GGC without explicit Laplace transform or
for HCM without closed expression for the density.
GGC and HCM densities
3
In 1981, Bondesson raised the conjecture that positive α−stable variables should be
HCM iff α ≤ 1/2 and this very hard problem (quoting his own recent words, see Remark
3 in Bondesson (2009)) is still unsolved except in the easy case when α is the reciprocal of
an integer - see Example 5.6.2 in Bondesson (1992). Notice, in passing, that the validity
of this conjecture is erroneously taken for granted in James, Roynette and Yor (2008)
p. 361. We refer to Bondesson (1981) pp. 54-55, Bondesson (1992) pp. 88-89 and also
to the manuscript Bondesson (1999) for several reasons, partly numerical, supporting
this hypothesis. Let us also mention the main theorem of Simon (2011), which states
that positive α−stable random variables are HM iff α ≤ 1/2. Actually, it follows easily
from the proofs of Lemmas 1 & 2 in Simon (2011) that the p-th power of a positive
(p/n)−stable variable is HCM for any integers p, n ≥ 2 such that p/n ≤ 1/2.
In the present paper we will present several examples of GGC and HCM densities
related to the above conjecture. In Section 2 we combine the main results of Roynette,
Vallois and Yor (2009) and Simon (2011) to show the GGC property for a large family of
negative powers of α−stable variables with α ≤ 1/4. This family is actually a bit larger
than the one which would be obtained from the validity of Bondesson’s hypothesis. The
more difficult case α ∈ (1/4, 1/2] is also studied, with a partial result. In Section 3,
we use Kanter’s and Pestana-Shanbhag-Sreehari’s factorizations to show that a large
class of positive powers of α−stable variables is the product of an HCM variable and an
ID variable. The latter turns out to be always a mixture of exponentials (ME), hence
very close to a GGC. Along the way, we also obtain an independent proof of PestanaShanbhag-Sreehari’s factorization. In Section 4 we show the aforementioned HCM result
for the quotient of two stable variables, and a similar characterization for Mittag-Leffler
variables. Not surprisingly, both yield the same boundary parameter α = 1/2.
The results presented in Section 2 and 3 are probably not optimal and at the end
of Section 3 we state another conjecture, where the power exponent α/(1 − α) appears
naturally. We also hope that the different tools and methods presented here will be helpful
to tackle Bondesson’s conjecture more deeply, even though we have tried to exploit them
to their full extent.
Notations. We will consider real random variables X having a density always denoted
by fX , unless explicitly stated. For the sake of brevity, we will use slightly incorrect
expressions like ”GGC variable” or ”HCM variable” and sometimes even delete the word
”variable” (as was actually already done in the present introduction). We will also set
”positive (negative) α-stable power” for ”positive (negative) power transformation of a
positive α−stable random variable”.
4
2. Negative α−stable powers and the GGC property
2.1. Some consequences of the HM property
Let Zα be a positive α−stable random variable - α ∈ (0, 1) - with density function fα
normalized such that
Z ∞
α
e−λt fα (t)dt = E e−λZα = e−λ , λ ≥ 0.
0
In the remainder of this paper we will use the notation β = 1 − α. We will also set Z1 = 1
by continuity. Recall that when α = 1/2, our normalization yields
f1/2 (x) =
1
√
e−1/4x 1{x>0} ·
2 πx3/2
(2.1)
Kanter’s factorization - see Corollary 4.1. in Kanter (1975) - reads
d
Zα = L−β/α × b−1/α
(U ),
α
(2.2)
where L ∼ Exp(1), U ∼ Unif(0, π) independent of L, and
bα (u) = (sin u/ sin(αu))α (sin u/ sin(βu))β ,
u ∈ (0, π),
is a bounded, decreasing and concave function - see Lemma 1 in Simon (2012). Observe
that when α = 1/2, Kanter’s factorization is a particular instance of the so-called BetaGamma algebra - see e.g. Bondesson (1992) pp. 13-14. Indeed, one has
d
−2
4b−2
(U/2) = Beta−1 (1/2, 1/2)
1/2 (U ) = cos
d
and (2.1) entails 4Z1/2 = Gamma−1 (1/2, 1), so that (2.2) amounts when α = 1/2 to
d
Gamma (1/2, 1) = Beta (1/2, 1/2) × Gamma (1, 1).
Put together with Shanbhag-Sreehari’s classical factorization of the exponential law see e.g. Exercise 29.16 in Sato (1999), Kanter’s factorization also shows that for every
γ ≥ α/β the random variable Zα−γ is ME, viz. there exists a positive random variable
Uα,γ such that
d
Zα−γ = L × Uα,γ .
(2.3)
See e.g. Section 51.1 in Sato (1999) for more material on ME random variables. With
the help of the HM property, one has the following reinforcement.
Proposition 2.1. For every γ > 0, the random variable Zα−γ is ID (with a CM density)
iff γ ≥ α/β. Moreover, Zα−γ is SD for every α ≤ 1/2 and every γ ≥ α/β.
5
Proof. The factorization (2.3) together with Theorem 51.6 and Proposition 51.8 in Sato
(1999) show that Zα−γ is ID with a CM density if γ ≥ α/β. On the other hand, a change
of variable and Linnik’s asymptotic expansion - see e.g. (14.35) in Sato (1999) - yield
xα/βγ log fZα−γ (x) → κα,γ ∈ (−∞, 0)
as x → ∞ for every γ > 0. Hence, if γ < α/β, Theorem 26.1 in Sato (1999) - see also
Exercise 29.10 therein - entails that Zα−γ is not ID. When α ≤ 1/2, the main result in
Simon (2011) shows that Zα is HM, so that log(Zα−γ ) has a log-concave density. If in
addition γ ≥ α/β, we have just observed that Zα−γ has a CM density which is hence
decreasing and log-convex. This entails that when α ≤ 1/2 and γ ≥ α/β, the random
variable Zα−γ belongs to the class mentioned in Bondesson (1992) Remark VI p. 28, and
is SD.
The main result of this section shows that the SD property for Zα−γ can be refined
into GGC, in some cases. The proof also relies on the HM property.
Theorem 2.1.
The random variable Zα−γ is GGC for any α ∈ (0, 1/4], γ ≥ 4α.
Proof. Fix α ∈ (0, 1/4], γ ≥ 4α and set δ = 2α/γ ∈ (0, 1/2]. Bochner’s subordination
for stable subordinators - see e.g. Example 30.5 in Sato (1999) - yields the identity
d
−1
−2α
Zα−2α = cα (Z1/2
× Z2α
)
for some purposeless constant cα > 0. We hence need to show the GGC property for the
random variable
2α 1/δ
(4Z1/2 )−1 × Z2α
)
,
whose density is in view of (2.1), the multiplicative convolution formula, and a series of
standard changes of variable, expressed as
Z
δxδ−3/2 ∞ −xδ y
√
e
f2α (y 1/2α )y 1/2α−1/2 dy.
α 2π 0
Observe that since
Z ∞
Z
1/2α 1/2α−1/2
f2α (y
)y
dy = 2α
0
0
∞
√
2α π
f2α (z)z dz =
< +∞,
Γ(1 − α)
α
(see e.g. (25.5) in Sato (1999) for the second equality), the function Kα f2α√(y 1/2α )y 1/2α−1/2
is a probability density on R+ , where we have set Kα = Γ(1 − α)/2α π. Denoting by
Xα the corresponding random variable, we have to show that
x 7→ Kα,δ xδ−3/2 E[e−x
δ
Xα
]
6
√
is the density of a GGC, with Kα,δ = 2δ/Γ(1 − α). Since δ ≤ 1/2 < 3/2, we see from
Theorem 6.2 in Bondesson (1990) (and the Remark 6.1 thereafter) that this will be done
as soon as
δ
1/δ
E[e−x Xα ] = E[e−x(Zδ ×Xα ) ]
is, up to normalization, the density of a GGC. We will now obtain this property with the
help of Theorem 2 in Roynette, Vallois and Yor (2009). On the one hand, it is easy to
1/δ
see that all negative moments of Zδ and Xα are finite, so that the density of Zδ × Xα
fulfils (1.1) in Roynette, Vallois and Yor (2009). On the other hand, the main result of
Simon (2011) entails that Z2α is HM because 2α ≤ 1/2, so that the function
t 7→ f2α (et/2α )et/2α−t/2
is log-concave and Xα is HM as well. Also, Zδ is HM because δ ≤ 1/2. Since the HM
1/δ
property is stable by independent multiplication, this shows that Zδ × Xα is HM, in
other words that it belongs to the class C defined in Roynette, Vallois and Yor (2009) p.
183, and we can apply Theorem 2 therein to conclude the proof.
Remarks 2.1. (a) From (14.30) in Sato (1999) and a change of variable, one has
sup{u ; lim fZα−γ (x)/xu−1 = 0} = α/γ
x→0
for every γ > 0, so that (3.1.4) in Bondesson (1992) shows that under the assumptions
of Theorem 2.1, the GGC random variable Zα−γ has a finite Thorin measure whose total
mass is α/γ. In other words, there exists a non-negative random variable Gα,γ such that
Z ∞
−γ
dx
, λ ≥ 0.
E[e−λZα ] = exp − (α/γ)
(1 − e−λx )E[e−xGα,γ ]
x
0
It would be interesting to get more properties of the random variables Gα,γ .
(b) It is easily seen that the above proof remains unchanged (and is even shorter) if
we take δ = 1 viz. γ = 2α, so that Zα−2α is GGC as well for any α ∈ (0, 1/4]. In view of
Theorem 2.1 and the general conjecture made in Bondesson (2009) Remark 3 (ii), it is
plausible that Zα−γ is GGC for any α ∈ (0, 1/4], γ ≥ 2α.
2.2. A certain family of densities on R+ and a partial result
A drawback of Theorem 2.1 is that it only covers the range α ∈ (0, 1/4]. Indeed, with
the same subordination method one should expect to handle the range α ∈ (1/4, 1/2]
as well. Motivated by the key-properties (2.20) and (2.23) in the proof of Theorem 2 in
Roynette, Vallois and Yor (2009), let us define the class P of probability densities f on
(0, +∞) satisfying
f (x)f (c/x) ≥ f (1/x)f (cx)
(2.4)
7
for all x, c > 0 such that (x − 1)(c − 1) ≥ 0. With an abuse of notation, we shall say that
a random variable X with density f belongs to P if f ∈ P. If X ∈ P, then it is easy
to see that X γ ∈ P for any γ 6= 0. Besides, it follows from Roynette, Vallois and Yor
(2009) pp. 187-188 that HM ⊂ P. Notice also that P 6⊂ HM, as the following example
shows. Consider the independent quotient Tα = (Zα /Zα )α which has an explicit density
gα given by
sin πα
gα (x) =
πα(x2 + 2 cos(πα)x + 1)
(see e.g. Exercise 4.21 (3) in Chaumont and Yor (2003)). A computation yields
(πα)2
1
1
−
= (1 − c2 )(x − 1/x)(x + 1/x + 2 cos πα),
gα (cx)gα (1/x)
sin2 πα gα (x)gα (c/x)
which is clearly non-positive whenever (x − 1)(c − 1) ≥ 0, so that Tα ∈ P for any
α ∈ (0, 1). However, it is easy to show - see the proof of Corollary 5 below - that Tα
is HM iff α ≤ 1/2. However, we know from (ix) p. 68 in Bondesson (1992) that Tα is
never HCM since gα has two poles eiπα and e−iπα in C\(−∞, 0]. Notice also that the
variable T1/2 is SD but not GGC - see Diédhiou (1998) and the references therein. We will
come back to this example in Section 4. The following proposition makes the relationship
between HM and P more precise.
Proposition 2.2.
For any non-negative random variable X having a density, one has
X is HM ⇐⇒ cX ∈ P ∀ c > 0.
Proof. The direct part is easy since cX is HM for any c > 0 whenever X is HM. For
the indirect part, setting gX (t) = log fX (et ) for any t ∈ R, the fact that cX ∈ P for
any c > 0 shows that for any −∞ < a ≤ b ≤ c ≤ d < +∞ with b + c = a + d, one has
gX (b) + gX (c) ≥ gX (a) + gX (d), so that gX is concave.
Together with the above example, this proposition entails that P is not stable by
multiplication with positive constants. Since on the other hand P is clearly stable under
weak convergence and since any positive constant can be approximated by a sequence
of truncated gaussian variables which all belong to HM ⊂ P, the unstability of P w.r.t.
constant multiplication entails that P is not - contrary to HM - stable by independent
multiplication either, viz. there exist independent X, Y ∈ P such that X × Y ∈
/ P.
Let now Y be a non-negative random variable with a density of the form κx−a E[e−xX ]
for some a ≥ 0 and a non-negative random variable X with finite negative moments such
that c−1 X ∈ P for some c > 0. A perusal of the proof of Theorem 2 in Roynette,
Vallois and Yor (2009) - see especially (2.10), (2.20) and (2.23) therein - shows, together
with Theorem 6.2 in Bondesson (1990), that ϕY (x) = E[e−xY ] is such that Hc (w) =
ϕY (cv)ϕY (c/v) is CM in the variable w = v + 1/v. On the other hand, it is possible to
show the following intrinsic property of positive stable densities.
8
Theorem 2.2.
For every α ∈ (0, 1), there exists cα ≥ 0 such that cZα ∈ P ⇔ c ≥ cα .
Though it has independent interest, we prefer not giving the proof of this theorem since
it is quite long, relying on the single intersection property for Zα - see Theorem 4.1 in
Kanter (1975), an extended Yamazato property for fα which is displayed in (1.4) in Simon
(2011) and the discussion thereafter, and a detailed analysis. Notice from Proposition 2.2
and the main result in Simon (2011) that cα = 0 for any α ≤ 1/2, and that necessarily
cα > 0 when α > 1/2. Theorem 2.2 and a painless adaptation of the proof of Theorem
2.1 entail the following property of the variable Zα−2α .
Corollary 2.1. For every α ∈ (1/4, 1/2], there exists c̃α > 0 such that for every c ∈
[0, c̃α ], the function Hcα (w) = ϕα (cv)ϕα (c/v) is CM in the variable w = v + 1/v, where
−2α
ϕα (x) = E[e−xZα ], x ≥ 0.
If we could show that c̃α = +∞, then Theorem 6.1.1 in Bondesson (1992) would
entail that Zα−2α is GGC for every α ∈ (1/4, 1/2]. Notice from Proposition 2.1 that when
α > 1/2 the random variable Zα−2α is not ID (since then 2α < α/β), hence not GGC.
From Corollary 2.1 and the above Remark 1 (b), it is very plausible that Zα−γ is GGC
for any α ≤ 1/2, γ ≥ 2α. See also the general Conjecture 3 raised at the end of the next
section.
3. On the infinite divisibility of Kanter’s random
variable
In this section we will deal with positive α−stable powers. From the point of view of
the factorization (2.2), we need to study negative powers of L and positive powers of the
−1/α
random variable bα (U ), which will be referred to as Kanter’s variable subsequently.
The latter plays an important role in simulation - see Devroye (2009) where it is called
Zolotarev’s variable, although the original computation leading to (2.2) is due to Chernine
and Ibragimov as explained in Kanter (1975). It is interesting to remark that Kanter’s
variable also appears explicitly in the context of free stable laws - see Demni (2011) p.
138 and the references therein.
Negative powers of L are not completely well understood from the point of view of
infinite divisibility. It follows from (iv) in Bondesson (1992) that Ls is HCM for every s ≤
−1, but it is not even known whether Ls is ID or not for s ∈ (−1, 0) - see Steutel and Van
Harn (2003) p. 521. Here, we will rather focus on positive powers of Kanter’s variable. Let
us first notice that the above factorization (2.3) was also observed in Pestana, Shanbhag
and Sreehari (1977), Theorem 1, where it is actually shown that Uα,γ = exp(−Wα,γ ) for
some ID random variable Wα,γ . In this section we will investigate (2.3) more thoroughly
and give finer properties of the random variable
−1
Vα = Uα,α/β
= b−1/β
(U ).
α
9
We could actually consider any positive power of Kanter’s random variable, but the latter
choice is more convenient for our purposes because of the identities
d
Zααs/β = L−s × Vαs
(3.1)
for every s ∈ R. Our purpose is three-fold. First, we provide an alternative proof of
Theorem 1 in Pestana, Shanbhag and Sreehari (1977), with the improvement that each
positive power Vαs (in particular, Kanter’s variable itself) is ID with a log-convex density.
Second, we show that all Vαs are actually positively translated ME’s. Third, we study in
some detail the case α = 1/2 and propose a general conjecture which is, in some sense,
a reinforcement of Bondesson’s.
3.1. Another proof of Pestana-Shanbhag-Sreehari’s factorization
Let us consider the random variable
Wα = log(Vα ) = −(1/β) log(bα (U ))
d
and observe that Wα,γ = γα−1 Wα + log(Zγα ) for every γ ≥ α/β, with the notation
γα = α/βγ. Since log(Zγα ) is ID - see e.g. Exercise 29.16 and Proposition 15.5 in Sato
(1999), Theorem 1 in Pestana, Shanbhag and Sreehari (1977) follows as soon as Wα is
ID. In view of Theorem 51.2 in Sato (1999) and the fact (obvious from the definition
of bα ) that the support of Wα is unbounded on the right, this is a consequence of the
following, which we prove independently of Pestana, Shanbhag and Sreehari (1977).
Theorem 3.1.
The density of Wα is log-convex.
−1/α
This theorem entails that all positive powers of Kanter’s random variable bα (U )
are ID, as shown in the next corollary. In particular, Zαγ is the product of a HCM random
variable and an ID random variable for every γ ≥ α/β.
Corollary 3.1.
For every s > 0, the density of Vαs is decreasing and log-convex.
Proof. Suppose first that s = 1. Since the support of Wα is unbounded on the right,
the log-convexity of its density entails that it is also decreasing, so that the function
gα (t) = fVα (et ) is also decreasing and log-convex. Since log x is increasing and concave,
this shows that fVα (x) = gα (log x) is decreasing and log-convex. The general case s > 0
follows analogously in considering the variable sWα = log(Vαs ).
The proof of Theorem 3.1 relies on the following lemma.
Lemma 3.1.
The function hα = b00α bα /(b0α )2 is increasing on (0, π).
10
Proof. First, observe that b1/2 (u) = 2 cos(u/2), so that h1/2 (u) = − cot2 (u/2), an increasing function on (0, π). In the general case the proof is more involved. Since bα = bβ ,
it is enough to consider the case α < 1/2. Set Aγ (u) = γ cot(γu) − cot(u) for every
γ ∈ (0, 1), f = αAα + βAβ and g = f 0 − f 2 . One has b0α = −f bα and b00α = −gbα , so that
hα = 1 + (1/f )0 and we need to show that 1/f is strictly convex, in other words that
2(f 0 )2 − f f 00 = 2gf 0 − f g 0 > 0.
(3.2)
It is shown in Lemma 1 of Simon (2012) that g = αβ + h + k, with the further notations
h = αβ(Aα − Aβ )2 and k = 2(Aα − Aβ )(βAβ − αAα ). On the other hand, the eulerian
formula
X
1
1
+ 2z
π cot(πz) =
2
z
z − n2
n≥1
shows that
Aγ (πz) =
2(1 − γ 2 )z X
n2
π
(n2 − z 2 )(n2 − γ 2 z 2 )
n≥1
is a strictly absolutely monotonic function on (0, 1) (i.e. all its derivatives are positive)
for every γ ∈ (0, 1), so that f = αAα + βAβ is absolutely monotonic on (0, π), too. In
particular, since αβ > 0, (3.2) holds if
(2hf 0 − f h0 ) + (2kf 0 − f k 0 ) ≥ 0.
A further computation entails 2hf 0 −f h0 = 2(Aα −Aβ )(A0β Aα −Aβ A0α ) and 2hf 0 −f h0 =
2((βAβ − αAα ) − 2αβ(Aα − Aβ ))(A0β Aα − Aβ A0α ), so that we need to prove
2(A0β Aα − Aβ A0α )((α2 + β 2 )(Aα − Aβ ) + (βAβ − αAα )) ≥ 0.
The above eulerian formula entails readily that Aα − Aβ and βAβ − αAα are positive
(actually, absolutely monotonic) functions on (0, π) and we are finally reduced to prove
A0β Aα − A0α Aβ ≥ 0.
(3.3)
We found no direct argument for the non-negativity of the above Wronskian. Writing
Aβ
β cot(βu) − cot(u)
sin(αu)
β cos(βu) sin(u) − cos(u) sin(βu)
=
(u) =
Aα
α cot(αu) − cot(u)
sin(βu)
α cos(αu) sin(u) − cos(u) sin(αu)
for every u ∈ (0, π) shows that
Aβ
Aα
0
(πz)
=
=
1
((1 − β 2 )Aα − (1 − α2 )Aβ + Aα Aβ (Aα − Aβ ))(πz)
A2α
8(β 2 − α2 )(1 − α2 )(1 − β 2 )z 3
(Sα (z)Sβ (z)S(z) − π 2 Sα,β (z)/4)
π 3 A2α (πz)
11
for every z ∈ (0, 1), with the notations
Sα (z) =
X
n≥1
S(z) =
X
n≥1
X
n2
n2
,
S
(z)
=
,
β
(n2 − α2 z 2 )(n2 − z 2 )
(n2 − β 2 z 2 )(n2 − z 2 )
n≥1
X
n2
n2
,
S
(z)
=
·
α,β
(n2 − α2 z 2 )(n2 − β 2 z 2 )
(n2 − α2 z 2 )(n2 − β 2 z 2 )(n2 − z 2 )
n≥1
Since S(z) ≥ S(0) = π 2 /6, we see that (3.3) is true if
Sα (z)Sβ (z) ≥ 3Sα,β (z)/2
for every z ∈ (0, 1). The latter follows e.g. after isolating the first term in each series
and using the fact that π 2 /6 ≥ 3/2. We leave the details to the reader. Observe that the
latter is also true for z = 0 because Sα (0) = Sβ (0) = π 2 /6 and Sα,β (0) = π 4 /90.
Proof of Theorem 3.1. We need to show that the density of log(b−1
α (U )) is log-convex,
in other words that the function
xfb0−1 (U ) (x)
α
(x)
fb−1
α (U )
0
is increasing over its domain of definition which is [αα β β , +∞). Since (log(b−1
α )) = f is
−1
a strictly absolutely monotonic function, the same holds for bα and we set b̃α for its
increasing reciprocal function. A computation yields
xfb0−1 (U ) (x)
xb̃00α (x)
=
= −2 +
(x)
fb−1
b̃0α (x)
α (U )
α
bα b00α
(b0α )2
(b̃α (x)),
which is an increasing function by Lemma 3.1.
3.2. Further properties of Wα and Vα
In Pestana, Shanbhag and Sreehari (1977), the infinite divisibility of Wα is proved together with a closed formula for its Lévy measure. In this paragraph, we use the latter
expression to show that Wα is actually a translated ME.
Theorem 3.2.
The density of Wα is CM.
As a consequence we obtain the following reinforcement of Corollary 3.1, which entails
that Zαγ is the product of a HCM random variable and a positively translated ME random
variable for every γ ≥ α/β :
12
Corollary 3.2.
For every s > 0, there exists cα,s > 0 such that Vαs − cα,s is ME.
Proof. Suppose first that s = 1. Theorem 3.2 shows that gα (t) = fVα (et ) is CM, with
the notation of Corollary 3.1. Since log x has a CM derivative, the classical Criterion 2 in
Feller (1953) p. 417 entails that fVα (x) = gα (log x) is CM over its domain of definition.
It is clear from the definition of bα that the latter is [βαα/β , +∞), so that Vα − βαα/β
is ME, by Proposition 51.8 in Sato (1999). The general case s > 0 follows analogously in
considering the density of sWα instead of Wα .
The proof of Theorem 3.2 relies on the following lemma.
Lemma 3.2.
The Lévy measure of Wα has a CM density given by
Z ∞
wα (x) = β
e−βtx ([t] − [αt] − [βt])dt, x ≥ 0,
1
where [t] stands for the integer part of any t ≥ 0.
Proof. From (3.1) and (3.5), (3.6) and (3.7) in Pestana, Shanbhag and Sreehari (1977)
- beware our notation β = 1 − α, for every λ ≥ 0 one has
E[e−βλWα ]
E[Zα−λα ]/E[Lλβ ]
Γ(1 + λ)
=
Γ(1 + λα)Γ(1 + λβ)
Z ∞
dx
= exp − aα λ +
(1 − e−λx − λx)gα (x)
x
0
=
for some constant aα ∈ R, where
gα (x) =
e−x
e−x/α
e−x/β
−
−
−x
−x/α
1−e
1−e
1 − e−x/β
is a non-negative function - see Lemma 3 in Pestana, Shanbhag and Sreehari (1977) which is also integrable with
Z ∞
gα (x) dx = −(α log α + β log β).
0
Besides, gα (x) → 1 as x → 0 so that one can rewrite
Z ∞
dx
E[e−βλWα ] = exp − ãα λ +
(1 − e−λx )gα (x)
x
0
(3.4)
for every λ ≥ 0, where ãα = α log α + β log β is the left-extremity of the support of the
variable βWα (this shows that the above constant aα is actually zero). Observe that
Z ∞
∞
X
gα (x) =
(e−(k+1)x − e−(k+1)x/α − e−(k+1)x/β ) =
e−tx µα (dt)
k=0
1
13
where
µα =
∞
X
(δk − δk/α − δk/β )
k=1
is a signed Radon measure over R+ . Integrating by parts, we get
Z ∞
Z ∞
−tx
e−tx ([t] − [αt] − [βt])dt.
e µα ([1, t))dt = x
gα (x) = x
1
1
Putting everything together shows that the density of the Lévy measure of Wα is given
by
Z ∞
e−βtx ([t] − [αt] − [βt])dt, x ≥ 0.
wα (x) = β
1
Proof of Theorem 3.2. From Lemma 3.2 and Theorem 51.10 in Sato (1999), we already
know that Wα −ãα belongs to the class B, which is the closure of ME for weak convergence
and convolution. Moreover, one has
Z ∞
wα (x) =
e−tx θα (t)dt, x ≥ 0,
0
with the notation θα (t) = ([t/β] − [t] − [(α/β)t])1{t≥β} , and it is clear that
Z 1
dt
θα (t)
< ∞ and 0 ≤ θα (t) ≤ 1, t ≥ 0.
t
0
From Theorem 51.12 in Sato (1999), this shows that Wα − ãα belongs to the class ME
itself, as required.
Remarks 3.1. (a) In the terminology of Schilling, Song and Vondraček (2010), Lemma
3.2 shows that the Lévy exponent of the ID random variable Wα is, up to translation, a
complete Bernstein function. Using Theorems 6.10 and 7.3 in Schilling, Song and Vondraček (2010) and simple transformations leads to the same conclusion that the density
of Wα is CM.
(b) Corollary 3.2 and Theorem 51.10 in Sato (1999) show that for every s > 0 the
Lévy measure of the random variable Vαs has a density vα,s which is CM. We believe that
x 7→ xvα,s (x) is also CM, in other words, that Vαs is actually GGC for every s > 0. See
Conjecture 1 below.
(c) The above Radon measure µα is signed and not everywhere positive, and we see
from (3.4) that Wα is not the translation of a GGC random variable. This latter property
might have been helpful for a better understanding of the random variables Vαs , although
it is still a conjecture - see Comment (1) p. 101 in Bondesson (1992) - that the transformation x 7→ ex − 1 leaves the GGC property invariant.
14
3.3. The case α = 1/2 and three conjectures
Taking α = 1/2 in (3.1) yields the factorizations
s
s
= L−s × V1/2
Z1/2
for every s ∈ R, where V1/2 has the explicit density
fV1/2 (x) =
1
p
1{x>1/4} .
2πx x − 1/4
s
Since log V1/2 has a log-convex density, the random variables V1/2
are not HM (this fact
was already noticed in Kanter (1975), see the remark before Theorem 4.1 therein) and
s
in particular not HCM. However, the following proposition shows that V1/2
is GGC at
s
−s
least for s ∈ [1/2, 1]. We set Ys = V1/2 − 4 for every s > 0.
Proposition 3.1.
The random variable Ys is HCM if and only if s ∈ [1/2, 1].
Proof. Changing the variable and using the fact that every function f on R+ is HCM
iff xλ f (1/x) is HCM for every λ ∈ R, one sees that the following equivalences hold:
Ys is HCM ⇔
1
(x + 1)
p
(x +
1)t
−1
is HCM ⇔
1
(x + 1)
p
(x + 1)t − xt
is HCM,
with the notation t = 1/s. Using the notation
ft (x) =
1
p
(x + 1) (x + 1)t − xt
for every t > 0, it is obvious that f1 and f2 are HCM. If now 1 < t < 2, then x 7→
(x+1)t −xt is obviously a Bernstein function (i.e.p
a positive function with CM derivative),
Criterion 2 in Feller (1953) p. 417 entails that 1/ (x + 1)t − xt is CM and ft is also CM.
Since ft (0) = 1, this shows that ft is the Laplace transform of some positive random
variable and, by Theorem 5.4.1 in Bondesson (1992), ft will be HCM iff the latter variable
is GGC. By the Pick function characterization given in Bondesson (1992) Theorem 3.1.2,
setting gt (z) = ft (−z) we need to show that g is analytic and zero-free on C\[0, ∞) and
Im (gt0 (z)/gt (z)) ≥ 0 for Im z > 0 (in other words, that gt0 /gt is a Pick function - see
Section 2.4 in Bondesson (1992)). The first point amounts to show that 1 − (z/(z − 1))t
does not vanish on C\[0, ∞), which is true because 1 < t < 2. For the second point, we
write
1
t
(1 − z)u − (−z)u
gt0 (z)
=
+
gt (z)
1−z
2 (1 − z)u+1 − (−z)u+1
with u = t − 1 ∈ (0, 1). Since 1/(1 − z) is obviously Pick, we need to show that
hu (z) = ((1 − z)u − (−z)u )((1 − z̄)u+1 − (−z̄)u+1 )
15
is also Pick for every u ∈ (0, 1). We compute
Im(hu (z))
=
Im(z)(|1 − z|2u + |z|2u ) − Im((−z)u (1 − z̄)u+1 + (−z̄)u+1 (1 − z)u )
=
Im(z)(|(1 − z)u − (−z̄)u |2 ) − Im((−z)u (1 − z̄)u ))
=
Im(z)(|(1 − z)u − (−z̄)u |2 ) + Im((|z|2 − z̄)u )
which is non-negative when Im z > 0 because u ∈ (0, 1). This shows the required property
and proves
p that Ys is HCM if s ∈ [1/2, 1]. Suppose now that s > 1 viz. t < 1. If
1/(x + 1) (1 + x)t − 1 were HCM, then it would also be HM and the function
y 7→ log(1 + ey ) +
1
log((1 + ey )t − 1)
2
would be convex on R. Differentiating the above entails that the function
t xt−1 − 1
t
1
+
∼ − t as x → ∞
x 7→
x
2
xt − 1
2x
would be non-increasing on [1, +∞), a contradiction. Last, in view of (ix) p. 6 in Bondesson (1992) it is easy to see that Ys is not HCM if s < 1/2, since (1 + z)t − 1 vanishes
at least twice on C\(−∞, 0] when t > 2. This completes the proof.
Remarks 3.2. (a) Except in the trivial cases t = 1 and t = 2, we could not find any
series of functions of the type described in (1.2) converging pointwise to ft when t ∈ [1, 2].
(b) From the above proposition, one might wonder if Kanter’s variable bα (U )−1/α is not
a translated HCM in general. If it were true, then (2.2) would show that Zα would be the
product of a HCM and a translated HCM for every α ≤ 1/2, so that from Theorem 6.2.2.
in Bondesson (1992) we would be quite close to the solution to Bondesson’s hypothesis.
Nevertheless, to show the above property raises computational difficulties significantly
greater than those in Lemma 1, and it does not seem that this approach could be simpler
than the one suggested in Bondesson (1992) pp. 88-89. See also Conjecture 1 below.
The following corollary shows that there are GGC random variables related to positive
α−stable powers with α ≥ 1/2.
Corollary 3.3.
For every α ∈ [1/2, 1), the random variable (Zα × Z1/2 )α is GGC.
Proof. Kanter’s and Shanbhag-Sreehari’s factorizations entail
d
d
α
(Zα × Z1/2 )α = (Zα /L)α × V1/2
= L−1 × (Yα + 4−α )
and from Proposition 3.1 and Theorem 4.3.1 in Bondesson (1992), we know that Yα +4−α
is GGC because α ∈ [1/2, 1). Since L−1 is HCM, Theorem 6.2.1 in Bondesson (1992)
shows that (Zα × Z1/2 )α is GGC as well.
16
s
As just mentioned, Proposition 3.1 shows that V1/2
is GGC for every s ∈ [1/2, 1]. From
the conjecture made in Bondesson (2009) Remark 3 (ii), one may ask if this property
does not remain true for every s > 1. Considering the variable Ys instead, this amounts
to show that the non-HCM function
x 7→
t
π(x + 1)
p
(x + 1)t − 1
is still a GGC density for every t ∈ (0, 1). From e.g. Example 15.2.2. in Schilling, Song
and Vondraček (2010), one sees that the Laplace transform of the renewal measure of
a tempered t−stable subordinator subordinated through a (1/2)-stable subordinator,
which is given by
1
,
x 7→ p
(x + 1)t − 1
is a factor in this density. However, we could not find any convenient expression for
the Stieltjes (i.e. double Laplace) transform of the above renewal measure which would
entail that the Laplace transform of Ys is HCM. If this renewal measure had a log-concave
density, we could apply Theorem 4.2.1. in Bondesson (1992), but this property does not
seem to be true even in the semi-explicit case t = 1/2 (inverse Gaussian distribution).
From the above Remark 3.1 (b) we know that for every s > 0 and every α ∈ (0, 1)
the Lévy measure of Vαs has a density vα,s which is CM (more precisely, of the form
given in Theorem 51.12 of Sato (1999)). Proposition 3.1 yields the reinforcement that
x 7→ xv1/2,s (x) is CM for every s ∈ [1/2, 1]. Eventhough this is a very particular case,
one might wonder if it is not true in general.
Conjecture 3.1.
The random variable Vαs is GGC for every s > 0 and every α ∈ (0, 1).
When α = 1/2, a part of this conjecture can be rephrased in terms of a more general
question on Beta variables. Since
d
(Beta (α1 , α2 ))−1 = 1 +
Gamma (α2 , 1)
Gamma (α1 , 1)
for every α1 , α2 > 0 - see e.g. Bondesson (1992) p.13, the HCM property for Gamma
variables and Theorem 5.1.1. in Bondesson (1992) show that (Beta (α1 , α2 ))−1 is always
a GGC. A particular case of the conjecture made in Bondesson (2009) Remark 3 (ii) is
hence the following.
Conjecture 3.2.
and every s ≥ 1.
The random variable (Beta (α1 , α2 ))−s is GGC for every α1 , α2 > 0
We could not find in the literature any result on the infinite divisibility of negative
powers of Beta variables. Proposition 3.1 shows that (Beta (1/2, 1/2))−s is a positively
translated HCM, hence GGC and ID, when s ∈ [1/2, 1]. The same conclusion holds for
17
other values of the parameters, not covering the full range α1 , α2 > 0. From the identities
(3.1) and Theorem 6.2.1 in Bondesson (1992), a positive answer to Conjecture 1 would
also show that Zαs is GGC for every s ≥ α/(1 − α). In view of our results in the previous
section for negative stable powers, it is tantalizing to raise the general
Conjecture 3.3.
α/(1 − α).
The random variable Zαγ is GGC for every α ∈ (0, 1) and |γ| ≥
If Bondesson’s HCM conjecture is true, then Zαs is GGC for every α ≤ 1/2 if |s| ≥ 1.
The above statement is stronger since it takes values of α which are greater than 1/2
in consideration, and since α/(1 − α) < 1 when α < 1/2. Notice that if Conjecture 3
is true, then from Proposition 2.1 we would also have Zα−γ is GGC ⇔ Zα−γ is ID for
every γ > 0. Last, since the main theorem of Simon (2012) shows that all positive stable
powers are unimodal, it would also be interesting to know if Zαγ is still a GGC when
γ ∈ (0, α/(1 − α)).
4. Related HCM densities
In this section we study two families of random variables which are related to Bondesson’s
hypothesis, and have their independent interest. For every α ∈ R, introduce the function
gα (u) =
1
u2α + 2 cos(πα) uα + 1
from R+ to R+ . The following elementary result might be well-known, although we could
not find any reference in the literature. As for Proposition 3.1, we could not find any
constructive argument either.
Proposition 4.1.
The function gα is HCM iff |α| ≤ 1/2.
Proof. Since u2α gα (u) HCM ⇔ gα (u) HCM and since cosine is an even function, it is
enough to consider the case α ≥ 0. Suppose first α > 1/2. Rewriting
gα (u) =
1
(uα + eiπα )(uα + e−iπα )
shows that gα has two poles in C/(−∞, 0] (because |(α − 1)/α| < 1) and from (ix) p. 68
in Bondesson (1992), this entails that gα is not HCM. Suppose next 0 ≤ α ≤ 1/2. The
cases α = 0 with m0 (u) = 1/4 and α = 1/2 with m1/2 (u) = 1/(u + 1) yield the HCM
property explicitly, and we only need to consider the case 0 < α < 1/2. Rewriting
gα (u) =
u−α
2 cos(πα) + u−α + uα
18
we see that it is enough to show that ρ : u 7→ 1/(c + uα + u−α ) is HCM for any c > 0.
Developing, one obtains
ρ(uv)ρ(u/v) =
1
c2
+
u2α
+
u−2α
+
v 2α
+
v −2α
+ c(uα + u−α )(v α + v −α )
for any u, v > 0. Since the function v 2α +v −2α +c(uα +u−α )(v α +v −α ) has CM derivative
in w = v + 1/v for any fixed u, c > 0 - see Bondesson (1990) p. 183, again Criterion 2 in
Feller (1953) p. 417 entails that the function ρ(uv)ρ(u/v) is CM in w, so that ρ is HCM.
This proposition has several interesting consequences. Let us first consider the random
variable
Yα = Tα1/α
with the notation of Section 2.2, and recall that it is the quotient of two independent
copies of Zα . The fact that Yα has a closed density seems to have been first noticed in
Lamperti (1958), in the context of occupation time for certain stochastic processes. If
Bondesson’s conjecture is true, then Yα is HCM whenever α ≤ 1/2, as a quotient of two
independent HCM random variables - see Theorem 5.1.1. in Bondesson (1992). The next
corollary shows that this is indeed the case.
Corollary 4.1.
The random variable Yα is HCM iff α ≤ 1/2.
Proof. From a fractional moment identification, the density of Yα is explicitly given by
fYα (x) =
π(x2α
sin πα α−1
sin παxα−1
=
x
gα (x)
+ 2xα cos πα + 1)
π
over R+ - see Exercise 4.21 (3) in Chaumont and Yor (2003) already mentioned in Section
2. The second derivative of t 7→ log fYα (et ) equals
−4α2 (1 + cos πα cosh αt)
(eαt + 2 cos πα + e−αt )2
and is not everywhere non-positive whenever α > 1/2, so that Yα is not HM, hence not
HCM either. When α ≤ 1/2, the above Proposition 4.1 shows immediately that fYα is a
HCM function, so that Yα is HCM as a random variable.
We now turn our attention to the so-called Mittag-Leffler random variables which were
introduced in Pillai (1990), and appeared since then in a variety of contexts. Let
Eα (x) =
∞
X
xn
Γ(1 + αn)
n=0
19
be the classical Mittag-Leffler function with index α ∈ (0, 1]. The Mittag-Leffler random
variable Mα has an explicit decreasing density given by
fMα (x) = αxα−1 Eα0 (−xα )
over R+ . Its Laplace transform is also explicit: for every λ ≥ 0 one has
Z ∞
1
−λx
α dx
−λMα
= exp −α
(1 − e
)Eα (−x )
] =
E[e
1 + λα
x
0
(4.1)
(see Remark 2.2 in Pillai (1990) and correct xk → xαk therein). From the classical fact
that x 7→ Eα (−x) is the Laplace transform of Zα−α - see e.g. Exercise 29.18 in Sato (1999)
- and hence a CM function, this shows that Mα is a GGC (with finite Thorin measure).
This latter fact follows also from the factorization
d
Mα = Zα × L1/α
where L ∼ Exp(1) (the latter is a direct consequence of the first equality in (4.1) - see
also the final remark in Pillai (1990)), and from Theorem 6.2.1. in Bondesson (1992)
since Zα is GGC and L1/α is HCM. Notice that in Example 3.2.4 of Bondesson (1990),
the GGC property for Mα is also obtained in a slightly more general context with the
help of Pick functions.
Corollary 4.2.
The random variable Mα is HCM iff α ≤ 1/2.
Proof. As a consequence of the above discussions, one has the classical representation
Z
h
i
α −α
uα−1 e−xu
sin πα
du
Eα (−xα ) = E e−x Zα
= E e−xYα =
2α + 2uα cos πα + 1
π
R+ u
so that the density of Mα writes
fMα (x) =
sin πα
π
Z
R+
u2α
uα e−xu
du.
+ 2uα cos πα + 1
From Proposition 4.1, the function
u 7→
π(u2α
sin πα uα
= ufYα (u)
+ 2uα cos πα + 1)
is HCM as soon as α ≤ 1/2, so that it is also a widened GGC density with the notations
of Section 3.5 in Bondesson (1990). By Theorem 5.4.1 in Bondesson (1992), this shows
that fMα is a HCM density function whenever α ≤ 1/2.
There are two ways to prove that Mα is not HCM for α > 1/2. First, again from
Theorem 5.4.1 in Bondesson (1990), it suffices to show that ufYα (u) is no more a widened
GGC density when α > 1/2. If it were true, then from Remark 6.1 in Bondesson (1990)
the function u1−δ fYα (u) would also be a widened GGC density for every δ > 0. In
20
particular, for every δ ∈]α, 1 + α[ the function u1−δ fYα (u) would be up to normalization
the density of a GGC. The derivative of the above function equals
−uα−δ−1 ((δ + α)u2α + 2δuα cos πα + (δ − α))
(u2α + 2uα cos πα + 1)2
and is easily seen to vanish twice on R+ if δ is close enough to α > 1/2. This shows that
the underlying variable is bimodal and contradicts Theorem 52.1 in Sato (1999) since all
GGC’s are SD.
The second argument shows that Mα is not even HM when α > 1/2. From (7) p. 207
in Erdélyi (1953) one has the asymptotic expansion
Eα (−z) =
N
−1
X
n=1
(−1)n+1 z −n
+ O(z −N ),
Γ(1 − αn)
z → +∞
(4.2)
for any N ≥ 2. Besides, by complete monotonicity, one can differentiate this expansion
term by term. Taking N = 3, one obtains
(zEα000 (−z) − Eα00 (−z))Eα0 (−z) − z(Eα00 (−z))2
=
−1
+ O(z −7 )
α2 z 6 Γ(−α)Γ(−2α)
as z → ∞, and the leading term in the right-hand side is positive when α > 1/2 : this
shows that t 7→ Eα0 (−et ) and, a fortiori, t 7→ αe(α−1)t Eα0 (−eαt ) are not log-concave, so
that Mα is not HM.
−α
Remarks 4.1. (a) Since Zα−α is not ID, the function Eα (−x) = E[e−xZα ] is CM but
never HCM. However, repeating verbatim the above argument shows that x 7→ Eα (−xα )
is HCM if and only if α ≤ 1/2.
(b) From the above proof, one has the equivalence
Mα is HM ⇐⇒ Mα is HCM ⇐⇒ α ≤ 1/2.
The variable L1/α is always HCM, hence HM, and we know that Zα is HM ⇔ α ≤ 1/2.
Since the HM property is closed by independent multiplication, this also proves that Mα
is HM as soon as α ≤ 1/2. On the other hand, the influence of the HM variable L1/α in
the product L1/α × Zα is not important enough to make Mα HM when α > 1/2. From
the above equivalence, one can ask if the general identification
HM ∩ GGC = HCM
is true or not, and we could not find any counterexample. Such an identification would
show Bondesson’s conjecture in view of the main result of Simon (2011).
(c) When α ≤ 1/2, the variable Mαr is clearly HM, hence unimodal, for every r 6= 0.
On the other hand, when α > 1/2, it is possible to find some r 6= 0 such that Mαr is not
21
unimodal. Indeed, it follows easily from the expansion (4.2) that
fM −1/α (0+) =
α
−1
−1
, f 0 −1/α (0+) =
> 0
αΓ(−α) Mα
αΓ(−2α)
and fMαs (0+) = +∞ ∀ s < −1/α. By continuity of s 7→ fMαs (x) for every x > 0, this
shows that Mαs is not unimodal for some s < −1/α close enough to −1/α. Hence, we
have shown the further equivalence
Mα is HM ⇐⇒ Mαr is unimodal for every r 6= 0.
(4.3)
The same equivalence holds for Zα as a consequence of the main results of Simon (2011,
2012) and we can raise the following natural question: For which class of positive random
variables with density does the equivalence (4.3) hold true? It is easy to see that if X is
such that X r is unimodal for every r 6= 0, then X is absolutely continuous.
(d) Reasoning exactly as at the beginning of Paragraph 2.2 in Simon (2012), the decomposition
Mαs = (L × Lα−1 )s/α × b−s/α
(U )
α
and the concavity of bα show that Mαs is unimodal as soon as s ≥ −α, for every α ∈
(0, 1). One might ask whether Mas is also unimodal for every −1/α ≤ s < −α. Notice
that the above reasoning is not valid anymore, at least for α = 1/2 because br1/2 (U ) =
2r (cos(U/2))r is bimodal as soon as r > 1.
(e) Both random variables Yα and Mα appear as special instances of the Lamperti-type
laws which were introduced in James (2010), to which we refer for a thorough study. See
also the numerous references therein.
Acknowledgements
This research was partially supported by grants ANR-09-BLAN-0084-01 and CMCU10G1503. The authors would like to thank L. Bondesson for the interest he took in this
paper, and also for providing them the manuscript Bondesson (1999).
References
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22
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Further examples of GGC and HCM densities

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